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Nonorientable $ 4$-manifolds with fundamental group of order $ 2$

Authors: Ian Hambleton, Matthias Kreck and Peter Teichner
Journal: Trans. Amer. Math. Soc. 344 (1994), 649-665
MSC: Primary 57N13; Secondary 57Q20, 57R67
MathSciNet review: 1234481
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Abstract: In this paper we classify nonorientable topological closed 4-manifolds with fundamental group $ \mathbb{Z}/2$ up to homeomorphism. Our results give a complete list of such manifolds, and show how they can be distinguished by explicit invariants including characteristic numbers and the $ \eta $-invariant associated to a normal $ Pin^c$-structure by the spectral asymmetry of a certain Dirac operator. In contrast to the oriented case, there exist homotopy equivalent nonorientable topological 4-manifolds which are stably homeomorphic (after connected sum with $ {S^2} \times {S^2}$) but not homeomorphic.

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